Rushbrooke inequality

{{More sources needed|date=December 2009}} In statistical mechanics, the '''Rushbrooke inequality'''<!-- who is Rushbrooke? --> relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature ''T''.

Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as

:<math> f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N </math>

The magnetization ''M'' per site in the thermodynamic limit, depending on the external magnetic field ''H'' and temperature ''T'' is given by

:<math> M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) </math>

where <math> \sigma_i </math> is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively

:<math> \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T </math>

and

:<math> c_H = T \left( \frac{\partial S}{\partial T} \right)_H. </math>

Additionally,

:<math> c_M = +T \left( \frac{\partial S}{\partial T} \right)_M. </math>

==Definitions== The critical exponents <math> \alpha, \alpha', \beta, \gamma, \gamma' </math> and <math> \delta </math> are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

:<math> M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 </math>

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:<math> M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 </math>

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:<math> \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} </math>

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:<math> c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} </math>

where

:<math> t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c}</math>

measures the temperature relative to the critical point.

==Derivation== Using the magnetic analogue of the Maxwell relations for the response functions, the relation

:<math> \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 </math>

follows, and with thermodynamic stability requiring that <math> c_H, c_M\mbox{ and }\chi_T \geq 0 </math>, one has

:<math> c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 </math>

which, under the conditions <math> H=0, t>0</math> and the definition of the critical exponents gives

:<math> (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} </math>

which gives the '''Rushbrooke inequality'''<ref>{{Cite book |last=Patterson |first=James |url=https://www.google.se/books/edition/Solid_State_Physics/YvuxwAR2RJAC?hl=en&gbpv=1&dq=%22Rushbrooke+inequality%22+-wikipedia&pg=PA419&printsec=frontcover |title=Solid-State Physics: Introduction to the Theory |last2=Bailey |first2=Bernard |date=2010-12-08 |publisher=Springer Science & Business Media |isbn=978-3-642-02589-1 |pages=419 |language=en}}</ref>

:<math> \alpha' + 2\beta + \gamma' \geq 2. </math>

Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.

==References== {{Reflist}}

{{DEFAULTSORT:Rushbrooke Inequality}} Category:Critical phenomena Category:Statistical mechanics